import Mathlib import OAI.Probability.SKGap.Model namespace OAI section noncomputable section open MeasureTheory ProbabilityTheory InformationTheory Real Set Filter open scoped NNReal ENNReal Topology noncomputable section open Real Set namespace SKGap def entropyEll (u : ℝ) : ℝ := -log (1 - u) - u def entropySlack (B u : ℝ) : ℝ := u ^ 2 / 2 - B * entropyEll u theorem entropyEll_hasDerivAt {u : ℝ} (hu : u < 1) : HasDerivAt entropyEll (u / (1 - u)) u := by have hd := ((((hasDerivAt_id u).const_sub 1).log (by linarith : 1 - u ≠ 0)).neg).sub (hasDerivAt_id u) convert! hd using 1 dsimp field_simp [show 1 - u ≠ 0 by linarith] ring theorem entropyEll_nonneg {u : ℝ} (hu : u < 1) : 0 ≤ entropyEll u := by have h := log_le_sub_one_of_pos (by linarith : 0 < 1 - u) dsimp [entropyEll] linarith theorem entropyEll_le_half_sq_of_nonpos {u : ℝ} (hu : u ≤ 0) : entropyEll u ≤ u ^ 2 / 2 := by let f : ℝ → ℝ := fun x => x ^ 2 / 2 - entropyEll x have hd (x : ℝ) (hx : x < 1) : HasDerivAt f (-x ^ 2 / (1 - x)) x := by convert! (((hasDerivAt_id x).pow 2).div_const 2).sub (entropyEll_hasDerivAt hx) using 1 dsimp field_simp [show 1 - x ≠ 0 by linarith] ring have hm : AntitoneOn f (Iic 0) := by apply antitoneOn_of_deriv_nonpos (convex_Iic 0) · intro x hx exact (hd x (by simp only [mem_Iic] at hx; linarith)).continuousAt.continuousWithinAt · intro x hx have hx0 : x < 0 := by simpa only [interior_Iic, mem_Iio] using hx exact (hd x (by linarith)).differentiableAt.differentiableWithinAt · intro x hx have hx0 : x < 0 := by simpa only [interior_Iic, mem_Iio] using hx rw [(hd x (by linarith)).deriv] exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by linarith) have h := hm (show u ∈ Iic 0 from hu) (show (0 : ℝ) ∈ Iic 0 by simp) hu simpa [f, entropyEll] using h theorem entropySlack_pos_of_neg {B u : ℝ} (hB0 : 0 ≤ B) (hB1 : B < 1) (hu : u < 0) : 0 < entropySlack B u := by have he := mul_le_mul_of_nonneg_left (entropyEll_le_half_sq_of_nonpos hu.le) hB0 have hs : 0 < u ^ 2 := sq_pos_of_ne_zero hu.ne dsimp [entropySlack] nlinarith theorem value_le_of_deriv_nonneg {f g : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (hd : ∀ x ∈ Icc a b, HasDerivAt f (g x) x) (hg : ∀ x ∈ Icc a b, 0 ≤ g x) : f a ≤ f b := by have hm : MonotoneOn f (Icc a b) := by apply monotoneOn_of_deriv_nonneg (convex_Icc a b) · intro x hx exact (hd x hx).continuousAt.continuousWithinAt · intro x hx exact (hd x (interior_subset hx)).differentiableAt.differentiableWithinAt · intro x hx rw [(hd x (interior_subset hx)).deriv] exact hg x (interior_subset hx) exact hm ⟨le_rfl, hab⟩ ⟨hab, le_rfl⟩ hab theorem half_sq_le_entropyEll {u : ℝ} (hu0 : 0 ≤ u) (hu1 : u < 1) : u ^ 2 / 2 ≤ entropyEll u := by let f : ℝ → ℝ := fun x => entropyEll x - x ^ 2 / 2 have hd (x : ℝ) (hx : x < 1) : HasDerivAt f (x ^ 2 / (1 - x)) x := by convert! (entropyEll_hasDerivAt hx).sub (((hasDerivAt_id x).pow 2).div_const 2) using 1 dsimp field_simp [show 1 - x ≠ 0 by linarith] ring have h := value_le_of_deriv_nonneg hu0 (fun x hx => hd x (hx.2.trans_lt hu1)) (fun x hx => div_nonneg (sq_nonneg x) (by linarith [hx.2])) simpa [f, entropyEll] using h theorem entropySlack_cubic_lower {u : ℝ} (hu0 : 0 ≤ u) (hu1 : u < 1) : u ^ 3 / 6 ≤ entropySlack (1 - u) u := by let f : ℝ → ℝ := fun x => entropySlack (1 - x) x - x ^ 3 / 6 have hd (x : ℝ) (hx : x < 1) : HasDerivAt f (entropyEll x - x ^ 2 / 2) x := by convert! ((((hasDerivAt_id x).pow 2).div_const 2).sub (((hasDerivAt_id x).const_sub 1).mul (entropyEll_hasDerivAt hx))).sub (((hasDerivAt_id x).pow 3).div_const 6) using 1 dsimp field_simp [show 1 - x ≠ 0 by linarith] ring have h := value_le_of_deriv_nonneg hu0 (fun x hx => hd x (hx.2.trans_lt hu1)) (fun x hx => sub_nonneg.mpr (half_sq_le_entropyEll hx.1 (hx.2.trans_lt hu1))) simpa [f, entropySlack, entropyEll] using h theorem entropySlack_cubic_lower_of_le {B u : ℝ} (hu0 : 0 ≤ u) (hu1 : u < 1) (hB : B ≤ 1 - u) : u ^ 3 / 6 ≤ entropySlack B u := by have h := mul_le_mul_of_nonneg_right hB (entropyEll_nonneg hu1) have hl := entropySlack_cubic_lower hu0 hu1 dsimp [entropySlack] at * linarith theorem entropySlack_pos {B u : ℝ} (hB0 : 0 ≤ B) (hB1 : B < 1) (huB : u ≤ 1 - B) (hu1 : u < 1) (hu0 : u ≠ 0) : 0 < entropySlack B u := by rcases hu0.lt_or_gt with hneg | hpos · exact entropySlack_pos_of_neg hB0 hB1 hneg · have h := entropySlack_cubic_lower_of_le hpos.le hu1 (by linarith : B ≤ 1 - u) exact (div_pos (pow_pos hpos 3) (by norm_num : (0 : ℝ) < 6)).trans_le h theorem log_one_add_cubic_upper {x : ℝ} (hx : 0 ≤ x) : log (1 + x) ≤ x - x ^ 2 / 2 + x ^ 3 / 3 := by let f : ℝ → ℝ := fun y => y - y ^ 2 / 2 + y ^ 3 / 3 - log (1 + y) have hd (y : ℝ) (hy : 0 ≤ y) : HasDerivAt f (y ^ 3 / (1 + y)) y := by convert! (((hasDerivAt_id y).sub (((hasDerivAt_id y).pow 2).div_const 2)).add (((hasDerivAt_id y).pow 3).div_const 3)).sub (((hasDerivAt_id y).const_add 1).log (by linarith : 1 + y ≠ 0)) using 1 dsimp field_simp [show 1 + y ≠ 0 by linarith] ring have h := value_le_of_deriv_nonneg hx (fun y hy => hd y hy.1) (fun y hy => div_nonneg (pow_nonneg hy.1 3) (by linarith [hy.1])) simpa [f] using h def entropyVarianceSlack (k B C : ℝ) : ℝ := C ^ 2 / k - 1 - log (C ^ 2 / k) - (C - B) ^ 2 / (1 + k) def entropyVarianceSlope (k B C : ℝ) : ℝ := 2 * C / k - 2 / C - 2 * (C - B) / (1 + k) theorem entropyVarianceSlack_hasDerivAt {k B C : ℝ} (hk : 0 < k) (hC : 0 < C) : HasDerivAt (entropyVarianceSlack k B) (entropyVarianceSlope k B C) C := by have hden : C ^ 2 / k ≠ 0 := div_ne_zero (pow_ne_zero 2 hC.ne') hk.ne' convert! (((((hasDerivAt_id C).pow 2).div_const k).sub_const 1).sub ((((hasDerivAt_id C).pow 2).div_const k).log hden)).sub ((((hasDerivAt_id C).sub_const B).pow 2).div_const (1 + k)) using 1 dsimp [entropyVarianceSlope] field_simp theorem entropyVarianceSlope_hasDerivAt {k B C : ℝ} (hk : 0 < k) (hC : 0 < C) : HasDerivAt (entropyVarianceSlope k B) (2 / (k * (1 + k)) + 2 / C ^ 2) C := by convert! ((((hasDerivAt_id C).const_mul 2).div_const k).sub ((hasDerivAt_const C (2 : ℝ)).div (hasDerivAt_id C) hC.ne')).sub ((((hasDerivAt_id C).sub_const B).const_mul 2).div_const (1 + k)) using 1 dsimp field_simp ring theorem entropyVariance_curvature_one {k C : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (_hC : 0 < C) : 1 ≤ 2 / (k * (1 + k)) + 2 / C ^ 2 := by have hp : 0 < k * (1 + k) := mul_pos hk0 (by linarith) have hk2 : k * (1 + k) ≤ 2 := by nlinarith have hdiv : 1 ≤ 2 / (k * (1 + k)) := by rwa [le_div_iff₀ hp, one_mul] have hpos : 0 ≤ 2 / C ^ 2 := div_nonneg (by norm_num) (sq_nonneg _) linarith theorem entropyVariance_taylor_lower {k B C₀ C : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (hC₀ : 0 < C₀) (hC : C₀ ≤ C) : entropyVarianceSlack k B C₀ + entropyVarianceSlope k B C₀ * (C - C₀) + (C - C₀)^2 / 2 ≤ entropyVarianceSlack k B C := by have hslope (x : ℝ) (hx : C₀ ≤ x) : entropyVarianceSlope k B C₀ + (x - C₀) ≤ entropyVarianceSlope k B x := by have h := value_le_of_deriv_nonneg hx (f := fun y => entropyVarianceSlope k B y - y) (g := fun y => 2 / (k * (1 + k)) + 2 / y ^ 2 - 1) (fun y hy => (entropyVarianceSlope_hasDerivAt hk0 (hC₀.trans_le hy.1)).sub (hasDerivAt_id y)) (fun y hy => sub_nonneg.mpr (entropyVariance_curvature_one hk0 hk1 (hC₀.trans_le hy.1))) linarith have hd (x : ℝ) (hx : C₀ ≤ x) : HasDerivAt (fun y => entropyVarianceSlack k B y - entropyVarianceSlope k B C₀ * (y - C₀) - (y - C₀)^2 / 2) (entropyVarianceSlope k B x - entropyVarianceSlope k B C₀ - (x - C₀)) x := by convert! ((entropyVarianceSlack_hasDerivAt hk0 (hC₀.trans_le hx)).sub (((hasDerivAt_id x).sub_const C₀).const_mul (entropyVarianceSlope k B C₀))).sub ((((hasDerivAt_id x).sub_const C₀).pow 2).div_const 2) using 1 dsimp ring have h := value_le_of_deriv_nonneg hC (fun x hx => hd x hx.1) (fun x hx => by linarith [hslope x hx.1]) simp only [sub_self, mul_zero, zero_pow (by norm_num : 2 ≠ 0), zero_div, sub_zero] at h linarith theorem entropyVariance_at_base {k a : ℝ} (hk : 0 < k) (ha : 0 < a) : entropyVarianceSlack k (1 - a) (1 + k * a) = 1 / k - 1 + log k + 2 * a - a ^ 2 - 2 * log (1 + k * a) := by have hC : 0 < 1 + k * a := by positivity unfold entropyVarianceSlack rw [log_div (pow_ne_zero 2 hC.ne') hk.ne', log_pow] field_simp ring theorem entropyVariance_at_base_lower {k a : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (ha : 0 ≤ a) : -(2 * a ^ 3 / 3) ≤ entropyVarianceSlack k (1 - a) (1 + k * a) := by have hC : 0 < 1 + k * a := by positivity have heq : entropyVarianceSlack k (1 - a) (1 + k * a) = 1 / k - 1 + log k + 2 * a - a ^ 2 - 2 * log (1 + k * a) := by unfold entropyVarianceSlack rw [log_div (pow_ne_zero 2 hC.ne') hk0.ne', log_pow] field_simp ring rw [heq] have hlogk := one_sub_inv_le_log_of_pos hk0 have hlog := log_le_log hC (show 1 + k * a ≤ 1 + a by nlinarith) have hcubic := log_one_add_cubic_upper ha simp only [one_div] at * linarith theorem entropyVarianceSlope_at_base {k a : ℝ} (hk : 0 < k) (ha : 0 ≤ a) : entropyVarianceSlope k (1 - a) (1 + k * a) = 2 / k - 2 / (1 + k * a) := by unfold entropyVarianceSlope field_simp ring theorem entropyVarianceSlope_at_base_lower {k a : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (ha : 0 ≤ a) : 2 * a / (1 + a) ≤ entropyVarianceSlope k (1 - a) (1 + k * a) := by rw [entropyVarianceSlope_at_base hk0 ha] have hkp : 0 < 1 + k * a := by positivity have hap : 0 < 1 + a := by positivity have heq : 1 / k - 1 / (1 + k * a) - a / (1 + a) = (1 - k) * (1 + a + k * a ^ 2) / (k * (1 + k * a) * (1 + a)) := by field_simp ring have hnonneg : 0 ≤ 1 / k - 1 / (1 + k * a) - a / (1 + a) := by rw [heq] positivity have h2 := mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hnonneg ring_nf at h2 ⊢ linarith theorem entropyVariance_pos_large {k a C : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (ha0 : 0 < a) (ha1 : a < 1) (hC : 1 + k * a ≤ C) (hlarge : (1 + k) * a ^ 3 / 3 ≤ (C - (1 + k * a)) ^ 2) : 0 < entropyVarianceSlack k (1 - a) C := by let Δ := C - (1 + k * a) have hΔ : 0 ≤ Δ := sub_nonneg.mpr hC have hΔsq : a ^ 3 / 3 ≤ Δ ^ 2 := by dsimp [Δ] have hp := mul_nonneg hk0.le (pow_nonneg ha0.le 3) nlinarith have hΔlower : a ^ 2 / 2 < Δ := by by_contra h have hle : Δ ≤ a ^ 2 / 2 := le_of_not_gt h have hsq := mul_self_le_mul_self hΔ hle have hp := mul_pos (pow_pos ha0 3) (show 0 < 4 - 3 * a by linarith) nlinarith have hlinslope : a ≤ 2 * a / (1 + a) := by rw [le_div_iff₀ (show 0 < 1 + a by positivity)] nlinarith [mul_nonneg ha0.le (sub_nonneg.mpr ha1.le)] have hlinear : a ^ 3 / 2 < entropyVarianceSlope k (1 - a) (1 + k * a) * Δ := by have hprod := mul_le_mul_of_nonneg_right (hlinslope.trans (entropyVarianceSlope_at_base_lower hk0 hk1 ha0.le)) hΔ have hstrict := mul_lt_mul_of_pos_left hΔlower ha0 nlinarith have ht := entropyVariance_taylor_lower (B := 1 - a) hk0 hk1 (show 0 < 1 + k * a by positivity) hC have hb := entropyVariance_at_base_lower hk0 hk1 ha0.le change entropyVarianceSlack k (1 - a) (1 + k * a) + entropyVarianceSlope k (1 - a) (1 + k * a) * Δ + Δ ^ 2 / 2 ≤ _ at ht linarith theorem entropy_transport_complete_square {k B C u : ℝ} (hk : 0 < k) : u * (C - B) - k * u ^ 2 / 2 - B * entropyEll u - (C - B) ^ 2 / (2 * (1 + k)) = entropySlack B u - (C - B - (1 + k) * u) ^ 2 / (2 * (1 + k)) := by unfold entropySlack field_simp ring theorem entropy_scalar_optimization {k B C R : ℝ} (hk0 : 0 < k) (hk1 : k ≤ 1) (hB0 : 0 < B) (hB1 : B < 1) (hR : 0 ≤ R) (htransport : ∀ u < 1, u * (C - B) - k * u ^ 2 / 2 - B * entropyEll u ≤ R) (hvariance : 1 < C → C ^ 2 / k - 1 - log (C ^ 2 / k) ≤ 2 * R) : (C - B) ^ 2 / (2 * (1 + k)) ≤ R ∧ (C ≠ B → (C - B) ^ 2 / (2 * (1 + k)) < R) := by have hkp : 0 < 1 + k := by linarith have hstrict (hCB : C ≠ B) : (C - B) ^ 2 / (2 * (1 + k)) < R := by let a := 1 - B have ha0 : 0 < a := sub_pos.mpr hB1 have ha1 : a < 1 := by dsimp [a]; linarith by_cases hsmall : (C - B) / (1 + k) ≤ a · let u := (C - B) / (1 + k) have hu1 : u < 1 := hsmall.trans_lt ha1 have hu0 : u ≠ 0 := div_ne_zero (sub_ne_zero.mpr hCB) hkp.ne' have hs := entropySlack_pos hB0.le hB1 hsmall hu1 hu0 have ht := htransport u hu1 have he := entropy_transport_complete_square (B := B) (C := C) (u := u) hk0 have hzero : C - B - (1 + k) * u = 0 := by dsimp [u]; field_simp; ring rw [hzero] at he norm_num at he linarith · have hC : 1 + k * a < C := by have ht := (lt_div_iff₀ hkp).mp (lt_of_not_ge hsmall) dsimp [a] at * nlinarith by_cases hlarge : (1 + k) * a ^ 3 / 3 ≤ (C - (1 + k * a)) ^ 2 · have hf := entropyVariance_pos_large hk0 hk1 ha0 ha1 hC.le hlarge have hv := hvariance (by have hp := mul_pos hk0 ha0; linarith) have hB : 1 - a = B := by dsimp [a]; ring rw [hB] at hf unfold entropyVarianceSlack at hf have he : (C - B) ^ 2 / (1 + k) = 2 * ((C - B) ^ 2 / (2 * (1 + k))) := by field_simp rw [he] at hf linarith · have hs := entropySlack_cubic_lower ha0.le ha1 have hB : 1 - a = B := by dsimp [a]; ring rw [hB] at hs have ht := htransport a ha1 have he := entropy_transport_complete_square (B := B) (C := C) (u := a) hk0 have hdelta : C - B - (1 + k) * a = C - (1 + k * a) := by dsimp [a]; ring rw [hdelta] at he have hbound : (C - (1 + k * a)) ^ 2 / (2 * (1 + k)) < a ^ 3 / 6 := by rw [div_lt_iff₀ (by positivity : 0 < 2 * (1 + k))] linarith [lt_of_not_ge hlarge] linarith refine ⟨?_, hstrict⟩ by_cases hCB : C = B · simpa [hCB] using hR · exact (hstrict hCB).le theorem entropy_variance_function_mono {w v : ℝ} (hw : 1 ≤ w) (hv : w ≤ v) : w - 1 - log w ≤ v - 1 - log v := by apply value_le_of_deriv_nonneg hv (g := fun x => 1 - 1 / x) · intro x hx have hxp : 0 < x := lt_of_lt_of_le (by norm_num : (0 : ℝ) < 1) (hw.trans hx.1) convert! ((hasDerivAt_id x).sub_const 1).sub ((hasDerivAt_id x).log hxp.ne') using 1 · intro x hx have hx1 : 1 ≤ x := hw.trans hx.1 rw [sub_nonneg, div_le_one (by linarith : 0 < x)] exact hx1 theorem entropy_variance_scale {j w : ℝ} (hj0 : 0 < j) (hj1 : j ≤ 1) (hw : 1 ≤ w) : w - 1 - log w ≤ j * (w / j - 1 - log (w / j)) := by have hwp : 0 < w := lt_of_lt_of_le (by norm_num : (0 : ℝ) < 1) hw rw [log_div hwp.ne' hj0.ne'] have hlog := mul_le_mul_of_nonneg_left (one_sub_inv_le_log_of_pos hj0) hj0.le have hextra := mul_nonneg (sub_nonneg.mpr hj1) (log_nonneg hw) have heq : j * (1 - j⁻¹) = j - 1 := by field_simp rw [heq] at hlog have hcancel : j * (w / j) = w := mul_div_cancel₀ w hj0.ne' nlinarith theorem entropy_variance_scaled_lower {j k C v D : ℝ} (hj0 : 0 < j) (hj1 : j ≤ 1) (hk0 : 0 < k) (hk1 : k ≤ 1) (hC : 1 < C) (hcv : C ^ 2 / (j * k) ≤ v) (hD : (v - 1 - log v) / 2 ≤ D) : C ^ 2 / k - 1 - log (C ^ 2 / k) ≤ 2 * (j * D) := by have hw : 1 ≤ C ^ 2 / k := by rw [le_div_iff₀ hk0]; nlinarith have hwj : 1 ≤ C ^ 2 / (j * k) := by rw [le_div_iff₀ (mul_pos hj0 hk0)] nlinarith [mul_le_mul_of_nonneg_right hj1 hk0.le] have hm := entropy_variance_function_mono hwj hcv have hs := entropy_variance_scale hj0 hj1 hw have heq : C ^ 2 / k / j = C ^ 2 / (j * k) := by field_simp rw [heq] at hs have hmj := mul_le_mul_of_nonneg_left hm hj0.le have hdj := mul_le_mul_of_nonneg_left hD hj0.le linarith end SKGap end end end end OAI